SAT MATH: ADDITIONAL TOPIC IN MATH
In SAT only 6 of the 58 questions come from this section and this is the most challenging section of SAT. Thus I would recommend you to study this part after you have gained confidence in all other sections.
In other sections there were some topics missing like Geometry, Trigonometry etc. Those remaining parts are discussed in this section. With the help of this section you will be able to see the beauty of nature like never before, if you keep your mind open. Geometry and trigonometry act as base for all architectural creations whether natural or man made.
GEOMETRY
In this part we will be discussing about geometry and mainly focusing on lines, circles and triangles.
LINE - The shortest distance between 2 points. It has no ends whereas a line segment has 2 fixed ends.
A ray has 1 fixed end and 1 extending to infinity.
Let there be a line AB and C be it's midpoint then it means that
AC = BC and the segment is divided into 2 equal parts.
ANGLE
The space between 2 intersecting lines is called as angle. It's a figure formed by 2 rays.
There are 5 types of angles.
ACUTE ANGLE → Less Than 90 Degree
RIGHT ANGLE → Equal To 90 Degree
OBTUSE ANGLE → Between 90 Degree And 180 Degree
STRAIGHT LINE → Equal To 180 Degree
REFLEX ANGLE → Between 180 Degree and 360 Degree
If there is an angle at vertex A formed by Rays AB and AC then it will be denoted as ∠BAC.
COMPLIMENTARY ANGLES - If 2 angles add up to 90 degrees then they are known as complimentary angle.
SUPPLEMENTARY ANGLES - If 2 angles add up to 180 degrees them they are known as Supplementary angle.
PARALLEL LINES
Two lines who never meet each other are called to be parallel lines. If you place 1 line over another you will find that they have angle of 0 degree between them.
Let's discuss about angles formed when 2 parallel lines are intersected by 1 another.
A) VERTICAL ANGLES - Angles with common vertex but different side of lines. Here angle 1 & 3, 2 & 4, 5 & 7 and 6 & 8 are vertical angles.
All the vertical angles are equal.
angle 1 = 3, 2 = 4, 5 = 7, 6 = 8
B) ALTERNATE INTERIOR ANGLES - They generally make a Z. They are interior angles present on opposite sides of the line intersecting the parallel lines. like angle 3 & 5, 4 & 6.
Alternate interior angles are equal
Angle 3 = 5 and 4 = 6
C) ALTERNATE EXTERIOR ANGLE - Just like alternate interior angles with the difference that in it there are exterior angles instead of interior angles.
Alternate exterior angles are equal
Angle 1 = 7 and 2 = 8
D) CORRESPONDING ANGLES - In it the angles are present at similar position. EX - angle 1 & 5, 2 & 6, 3 & 7 and 4 & 8.
Corresponding angles are equal
Angle 1 = 5, 2 = 6, 3 = 7, 4 = 8
# EQUATION OF LINE
You see the slope = ΔY/ΔX
We will be using this formula when 2 points like (X1,Y1) and (X2,Y2) are given.
slope = (Y – Y1)/(X – X1)
= (Y2 – Y1)/(X2 – X1)
CIRCLE
A circle is a figure equidistant from 1 point, with that 1 point referred as it's center.
Area of circle = πR2
Here this values is for the complete circle if you want to find the values for a sector that is a section of a circle
then multiply it by θ/2π, where θ is the angle subtended by arc on the center.
So for a sector subtending angle θ at center.
Area of circle = πR2×θ/2π = R2θ/2
Circumference = 2πR × θ/2π = Rθ
The total angle a point can subtend is 360 degree. like here it is
180 degree + 180 degree .
The conversion of Radian and degree is
180 degree = π
360 degree = 2π
So when we multiply it by θ/2π we are actually multiplying this to get the share of that sector.
EQUATION OF CIRCLE
for a circle with center (X1,Y1) and radius r.
Your goal is to convert all the forms of equation of circle into this.
THEOREMS OF CIRCLE
1) Angle at the center is twice the angle at circumference.
2) Angle in semicircle is a right angle.
3) Angles in same segments are equal.
4) opposite angles in a cyclic quadrilateral sum up to 180 degree. Here, cyclic quadrilateral is a quadrilateral whose all 4 vertices lie on the circumference of a circle.
# TANGENTS AND NORMAL
Tangent - A line touching the circle at only 1 point. It's perpendicular to the radius.
Normal - The line joining the center of a circle to tangent is called as normal.
#CHORDS
The line joining any 2 segment of circle is called a chord. Diameter is a special chord which passes through center. It's the longest chord.
TRIANGLES
In this section we will be discussing about the various properties of triangles. We already know that a triangle is a figure of 3 sides and 3 angles.
Let's start with the Pythagorean theorem.
In right triangle where one of an angle is 90 degree
Hypotenuse = Side2
It will be very handy in questions asking length of a side with questions combining circles and triangles.
EQUILATERAL TRIANGLE - In these all the 3 sides are of equal length and all the angles are equal to 60 degrees.
ISOSCELES TRIANGLE - In these 2 sides are equal and their opposite angles are equal.
CONGRUENCE
Two figures are said to be congruent when they can be imposed over one another i.e. they are the exact copy of one another.
There are certain conditions for 2 triangles to be congruent i.e.
1)SSS (Side Side Side) - All sides of both triangles are equal.
2) SAS (Side Angle Side) - 2 Sides and angle between them is equal.
3) AAS (Angle Angle Side) - 2 Angles and side are equal
4) HL (Hypotenuse Leg) - For 2 right angled triangles their Hypotenuse and one side is equal.
SIMILARITY
Two figures are said to be similar if their shape is same but their size may differ.
Condition to find if 2 triangles are similar.
1) SSS(Side Side Side) - If all the corresponding sides are proportional.
2)SAS(Side Angle Side) - If 2 sides are proportional and angle between them is same.
3) AA(Angle Angle) - If 2 of the angles are equal.
If ΔABC is similar to ΔDEF
then, ∠A = ∠D,∠B = ∠E and ∠C = ∠F
AB/DE = BC/EF = AC/DF
TRIANGLE INEQUALITY THEOREM - Sum of the lengths of any 2 sides is greater than the length of third side.
REGULAR POLYGON - Any polygon with all sides equal in length is called as Regular Polygon.
SQUARE - A quadrilateral with all 4 sides equal and both diagonals equal. All 4 angles = 90 degrees
Perimeter = 4 × side
Area = side2
RECTANGLE - A quadrilateral with opposite sides equal and with both diagonals equal. All 4 angles = 90 degrees.
Perimeter = 2( length + breadth)
Area = length × breadth
PARALLELOGRAM - A quadrilateral with opposite sides equal and parallel. Diagonals bisect each other.
Perimeter = 2( length + breadth)
Area = length × breadth
TRAPEZIUM - A quadrilateral in which 1 pair of sides are parallel and remaining are not.
Perimeter = Sum of 4 sides
Area = height × (sum of parallel sides)/2
AREA AND VOLUME
In SAT, area and volume formulas are given in the question paper so you don't have to memorize it. You just need to apply them.
TRIGONOMETRY
In a right angled triangle ABC Let ∠A be θ
sinθ = opposite side/hypotenuse
cosθ = adjacent side/hypotenuse
tanθ = opposite side/adjacent side
it's also referred as Soh Cah Toa in short. You can remember it in this way.
cotθ = adjacent side/opposite side
secθ = hypotenuse/adjacent side
cosecθ = hypotenuse/opposite side
PROPERTIES
sin(90-θ) = cosecθ
cos(90-θ) = secθ
tan(90-θ) = cotθ
sinθ = 1/cosecθ
cosθ = 1/secθ
tanθ = 1/cotθ
These identities and properties are very handy while solving SAT questions.
Now come the less important ones
COMPLEX NUMBERS
Till now all the numbers we have studies were real numbers. Now come the complex or imaginary numbers. They contain an iota referred as 'i'.
Z(Complex number) = a +bi where i = √(-1)
conjugate of Z = a - bi
the property of conjugate is that in it, the imaginary part get's it's sign changed.
Now coming to iota as you can see
and i4 = 1
thus if you want to calculate i127 remove i124 as it's multiple of i4 thus equal to 1 and calculate
i3 = -1 ×i = -i
SAT HOMEPAGE
SAT EXAM FORMAT
SAT READING
SAT WRITING
SAT MATH
SAT MATH: HEART OF ALGEBRA
SAT MATH: PROBLEM SOLVING AND DATA ANALYSIS
SAT MATH: ADDITIONAL TOPIC IN MATH
SAT MATH: PASSPORT TO ADVANCE MATH
SAT MATH:GRAPH
SAT ESSAY
In other sections there were some topics missing like Geometry, Trigonometry etc. Those remaining parts are discussed in this section. With the help of this section you will be able to see the beauty of nature like never before, if you keep your mind open. Geometry and trigonometry act as base for all architectural creations whether natural or man made.
GEOMETRY
In this part we will be discussing about geometry and mainly focusing on lines, circles and triangles.
LINE - The shortest distance between 2 points. It has no ends whereas a line segment has 2 fixed ends.
A ray has 1 fixed end and 1 extending to infinity.
Let there be a line AB and C be it's midpoint then it means that
AC = BC and the segment is divided into 2 equal parts.
ANGLE
The space between 2 intersecting lines is called as angle. It's a figure formed by 2 rays.
There are 5 types of angles.
ACUTE ANGLE → Less Than 90 Degree
RIGHT ANGLE → Equal To 90 Degree
OBTUSE ANGLE → Between 90 Degree And 180 Degree
STRAIGHT LINE → Equal To 180 Degree
REFLEX ANGLE → Between 180 Degree and 360 Degree
If there is an angle at vertex A formed by Rays AB and AC then it will be denoted as ∠BAC.
COMPLIMENTARY ANGLES - If 2 angles add up to 90 degrees then they are known as complimentary angle.
SUPPLEMENTARY ANGLES - If 2 angles add up to 180 degrees them they are known as Supplementary angle.
PARALLEL LINES
Two lines who never meet each other are called to be parallel lines. If you place 1 line over another you will find that they have angle of 0 degree between them.
Let's discuss about angles formed when 2 parallel lines are intersected by 1 another.
A) VERTICAL ANGLES - Angles with common vertex but different side of lines. Here angle 1 & 3, 2 & 4, 5 & 7 and 6 & 8 are vertical angles.
All the vertical angles are equal.
angle 1 = 3, 2 = 4, 5 = 7, 6 = 8
B) ALTERNATE INTERIOR ANGLES - They generally make a Z. They are interior angles present on opposite sides of the line intersecting the parallel lines. like angle 3 & 5, 4 & 6.
Alternate interior angles are equal
Angle 3 = 5 and 4 = 6
C) ALTERNATE EXTERIOR ANGLE - Just like alternate interior angles with the difference that in it there are exterior angles instead of interior angles.
Alternate exterior angles are equal
Angle 1 = 7 and 2 = 8
D) CORRESPONDING ANGLES - In it the angles are present at similar position. EX - angle 1 & 5, 2 & 6, 3 & 7 and 4 & 8.
Corresponding angles are equal
Angle 1 = 5, 2 = 6, 3 = 7, 4 = 8
# EQUATION OF LINE
you can find the equation of a line if slope and a point is given.
Let slope = m, point = (X1,Y1)
m = (Y – Y1)/(X – X1)
You see the slope = ΔY/ΔX
We will be using this formula when 2 points like (X1,Y1) and (X2,Y2) are given.
slope = (Y – Y1)/(X – X1)
= (Y2 – Y1)/(X2 – X1)
CIRCLE
A circle is a figure equidistant from 1 point, with that 1 point referred as it's center.
Area of circle = πR2
Here this values is for the complete circle if you want to find the values for a sector that is a section of a circle
then multiply it by θ/2π, where θ is the angle subtended by arc on the center.
So for a sector subtending angle θ at center.
Area of circle = πR2×θ/2π = R2θ/2
Circumference = 2πR × θ/2π = Rθ
The total angle a point can subtend is 360 degree. like here it is
180 degree + 180 degree .
The conversion of Radian and degree is
180 degree = π
360 degree = 2π
So when we multiply it by θ/2π we are actually multiplying this to get the share of that sector.
EQUATION OF CIRCLE
for a circle with center (X1,Y1) and radius r.
(X - X1)2 + (Y – Y1)2
= R2
Your goal is to convert all the forms of equation of circle into this.
THEOREMS OF CIRCLE
1) Angle at the center is twice the angle at circumference.
2) Angle in semicircle is a right angle.
3) Angles in same segments are equal.
4) opposite angles in a cyclic quadrilateral sum up to 180 degree. Here, cyclic quadrilateral is a quadrilateral whose all 4 vertices lie on the circumference of a circle.
# TANGENTS AND NORMAL
Tangent - A line touching the circle at only 1 point. It's perpendicular to the radius.
Normal - The line joining the center of a circle to tangent is called as normal.
#CHORDS
The line joining any 2 segment of circle is called a chord. Diameter is a special chord which passes through center. It's the longest chord.
TRIANGLES
In this section we will be discussing about the various properties of triangles. We already know that a triangle is a figure of 3 sides and 3 angles.
Let's start with the Pythagorean theorem.
In right triangle where one of an angle is 90 degree
Hypotenuse = Side2
C2
= A2 + B2
It will be very handy in questions asking length of a side with questions combining circles and triangles.
EQUILATERAL TRIANGLE - In these all the 3 sides are of equal length and all the angles are equal to 60 degrees.
ISOSCELES TRIANGLE - In these 2 sides are equal and their opposite angles are equal.
CONGRUENCE
Two figures are said to be congruent when they can be imposed over one another i.e. they are the exact copy of one another.
There are certain conditions for 2 triangles to be congruent i.e.
1)SSS (Side Side Side) - All sides of both triangles are equal.
2) SAS (Side Angle Side) - 2 Sides and angle between them is equal.
3) AAS (Angle Angle Side) - 2 Angles and side are equal
4) HL (Hypotenuse Leg) - For 2 right angled triangles their Hypotenuse and one side is equal.
SIMILARITY
Two figures are said to be similar if their shape is same but their size may differ.
Condition to find if 2 triangles are similar.
1) SSS(Side Side Side) - If all the corresponding sides are proportional.
2)SAS(Side Angle Side) - If 2 sides are proportional and angle between them is same.
3) AA(Angle Angle) - If 2 of the angles are equal.
If ΔABC is similar to ΔDEF
then, ∠A = ∠D,∠B = ∠E and ∠C = ∠F
AB/DE = BC/EF = AC/DF
TRIANGLE INEQUALITY THEOREM - Sum of the lengths of any 2 sides is greater than the length of third side.
REGULAR POLYGON - Any polygon with all sides equal in length is called as Regular Polygon.
SQUARE - A quadrilateral with all 4 sides equal and both diagonals equal. All 4 angles = 90 degrees
Perimeter = 4 × side
Area = side2
RECTANGLE - A quadrilateral with opposite sides equal and with both diagonals equal. All 4 angles = 90 degrees.
Perimeter = 2( length + breadth)
Area = length × breadth
PARALLELOGRAM - A quadrilateral with opposite sides equal and parallel. Diagonals bisect each other.
Perimeter = 2( length + breadth)
Area = length × breadth
TRAPEZIUM - A quadrilateral in which 1 pair of sides are parallel and remaining are not.
Perimeter = Sum of 4 sides
Area = height × (sum of parallel sides)/2
AREA AND VOLUME
In SAT, area and volume formulas are given in the question paper so you don't have to memorize it. You just need to apply them.
TRIGONOMETRY
In a right angled triangle ABC Let ∠A be θ
sinθ = opposite side/hypotenuse
cosθ = adjacent side/hypotenuse
tanθ = opposite side/adjacent side
it's also referred as Soh Cah Toa in short. You can remember it in this way.
cotθ = adjacent side/opposite side
secθ = hypotenuse/adjacent side
cosecθ = hypotenuse/opposite side
PROPERTIES
sin(90-θ) = cosecθ
cos(90-θ) = secθ
tan(90-θ) = cotθ
cot(90-θ) = tanθ
sec(90-θ) = cosθ
cosec(90-θ) = sinθ
sec(90-θ) = cosθ
cosec(90-θ) = sinθ
sinθ = 1/cosecθ
cosθ = 1/secθ
tanθ = 1/cotθ
cotθ = 1/tanθ
secθ = 1/cosθ
cosecθ = 1/ sinθ
secθ = 1/cosθ
cosecθ = 1/ sinθ
Sin2θ + cos2θ = 1
Sec2θ
= 1 + tan2θ
Cosec2θ
= 1 + cot2θ
These identities and properties are very handy while solving SAT questions.
Now come the less important ones
sin(a+b) = sin(a)cos(b)
+ cos(a)sin(b)
sin(a-b) =
sin(a)cos(b) - cos(a)sin(b)
cos(a+b) =
cos(a)cos(b) – sin(a)sin(b)
cos(a-b) =
cos(a)cos(b) + sin(a)sin(b)
COMPLEX NUMBERS
Till now all the numbers we have studies were real numbers. Now come the complex or imaginary numbers. They contain an iota referred as 'i'.
Z(Complex number) = a +bi where i = √(-1)
conjugate of Z = a - bi
the property of conjugate is that in it, the imaginary part get's it's sign changed.
i2 = -1
and i4 = 1
thus if you want to calculate i127 remove i124 as it's multiple of i4 thus equal to 1 and calculate
i3 = -1 ×i = -i
BONNE CHANCE !
SAT HOMEPAGE
SAT EXAM FORMAT
SAT READING
SAT WRITING
SAT MATH
SAT MATH: HEART OF ALGEBRA
SAT MATH: PROBLEM SOLVING AND DATA ANALYSIS
SAT MATH: ADDITIONAL TOPIC IN MATH
SAT MATH: PASSPORT TO ADVANCE MATH
SAT MATH:GRAPH
SAT ESSAY
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